10 - Convergence of Complex Sequences

Convergence of Complex Series

Let $(z_n)$ be a complex sequence
Represent $z_n=x_n+i{y}_n$ with $x_n,y_n\in \mathbb{R}$ so $(x_n)$ and $(y_n)$ are real sequences
Then $(z_n)$ converges if and only iff

$$(x_n)\text{ and }(y_n)\text{ both converge}$$

Moreover if $(z_n)$ converges then

$$\underset{n\to \infty }{\lim }{z}_n=\underset{n\to \infty }{\lim }{x}_n+i\underset{n\to \infty }{\lim }{y}_n$$